论文标题
关于拓扑复杂性的增长
On the growth of topological complexity
论文作者
论文摘要
令$ \ mathrm {tc} _r（x）$表示$ r $ - 空间$ x $的拓扑复杂性。在许多情况下，生成函数$ \ sum_ {r \ ge 1} \ mathrm {tc} _ {r+1}（x）（x）x^r $是一个有理函数$ \ frac {p（x）} {（x）{（1-x）^2} $ p（x）$ p（x）$ p（x）went $ \ mathrm {tc} _r（x）$相对于$ r $的渐近生长是$ \ mathrm {cat}（x）$。在本文中，我们引入了一个$ \ mathrm {tc} _rm {tc} _r（x）$的下限$ \ mathrm {mtc} _r（x）$ of Ricational Space $ X $，并估计$ \ Mathrm {mtc} _r（x）$的增长。
Let $\mathrm{TC}_r(X)$ denote the $r$-th topological complexity of a space $X$. In many cases, the generating function $\sum_{r\ge 1}\mathrm{TC}_{r+1}(X)x^r$ is a rational function $\frac{P(x)}{(1-x)^2}$ where $P(x)$ is a polynomial with $P(1)=\mathrm{cat}(X)$, that is, the asymptotic growth of $\mathrm{TC}_r(X)$ with respect to $r$ is $\mathrm{cat}(X)$. In this paper, we introduce a lower bound $\mathrm{MTC}_r(X)$ of $\mathrm{TC}_r(X)$ for a rational space $X$, and estimate the growth of $\mathrm{MTC}_r(X)$.
